Clusters of repetition roots

在过去的一年里，我们成功地将锚点位置技术扩展到表示指数大于2的重复。利用这一点，我们证明了我们的簇大小猜想对于具有任意整数指数的重复根的单链成立。我们还证明了所给出的单链的界是最优的，因为一个构造性证明给出了允许的簇大小的任意组合的簇序列。这些结果已经发表在DCFS 2022上。去年的一个大发展是Li和Brlek利用Rauzy图的性质解决了Fraenkel-Simpson猜想。我们试图将他们的论点推广到所谓的扩展Rauzy图，以在一般情况下证明我们的簇猜想。到目前为止，介绍这些结果的论文正在编写中。

In the last year we managed to extend the technique of anchor positions to represent repetitions with exponents higher than 2. Using this, we showed that our cluster size conjecture holds for single chains of repetition roots with arbitrary integer exponent.We also showed that the bounds given for single chains are optimal by a constructive proof yielding sequences of clusters for any combination of cluster sizes allowed within the bounds. These results have been published in DCFS 2022.A big development from last year was the solution of the Fraenkel-Simpson conjecture by Li and Brlek using properties of Rauzy graphs. We tried to extend their arguments to so called extended Rauzy graphs to prove our clusters conjecture in the general case. The paper presenting those results is under work as of now.